CHAPTER TWENTY-FOUR

PORTFOLIO PERFORMANCE

EVALUATION

MEASURES OF RETURN

MEASURES OF RETURN•complicated by addition or withdrawal

of money by the investor

•percentage change is not reliable when the base amount may be changing

•timing of additions or withdrawals is important to measurement

MEASURES OF RETURN

TWO MEASURES OF RETURN•Dollar-Weighted Returns

uses discounted cash flow approachweighted because the period with the

greater number of shares has a greater influence on the overall average

MEASURES OF RETURN

TWO MEASURES OF RETURN•Time-Weighted Returns

used when cash flows occur between beginning and ending of investment horizon

ignores number of shares held in each period

MEASURES OF RETURN

TWO MEASURES OF RETURN•Comparison of Time-Weighted to

Dollar-Weighted ReturnsTime-weighted useful in pension fund

management where manager cannot control the deposits or withdrawals to the fund

MAKING RELEVANT COMPARISONS PERFORMANCE

•should be evaluated on the basis of a relative and not an absolute basisthis is done by use of a benchmark

portfolio

•BENCHMARK PORTFOLIOshould be relevant and feasiblereflects objectives of the fundreflects return as well as risk

THE USE OF MARKET INDICES INDICES

•are used to indicate performance but depend uponthe securities used to calculate themthe calculation weighting measures

THE USE OF MARKET INDICES INDICES

•Three Calculation Weighting Methods:price weighting

– sum prices and divided by a constant to determine average price

– EXAMPLE: THE DOW JONES INDICES

THE USE OF MARKET INDICES INDICES

•Three Calculation Weighting Methods:value weighting (capitalization method)

– price times number of shares outstanding is summed

– divide by beginning value of index– EXAMPLE:

• S&P500• WILSHIRE 5000• RUSSELL 1000

THE USE OF MARKET INDICES INDICES

•Three Calculation Weighting Methods:equal weighting

– multiply the level of the index on the previous day by the arithmetic mean of the daily price relatives

– EXAMPLE:• VALUE LINE COMPOSITE

ARITHMETIC V. GEOMETRIC AVERAGES GEOMETRIC MEAN FRAMEWORK

GM = ( HPR)1/N - 1where = the summation of the

product of HPR= the holding period returns n= the number of periods

ARITHMETIC V. GEOMETRIC AVERAGES GEOMETRIC MEAN FRAMEWORK

•measures past performance well

•represents exactly the constant rate of return needed to earn in each year to match some historical performance

ARITHMETIC V. GEOMETRIC AVERAGES ARITHMETIC MEAN FRAMEWORK

•provides a good indication of the expected rate of return for an investment during a future individual year

•it is biased upward if you attempt to measure an asset’s long-run performance

RISK-ADJUSTED MEASURES OF PERFORMANCE THE REWARD TO VOLATILITLY

RATIO (TREYNOR MEASURE)•There are two components of risk

risk associated with market fluctuationsrisk associated with the stock

•Characteristic Line (ex post security line)defines the relationship between historical

portfolio returns and the market portfolio

TREYNOR MEASURE

TREYNOR MEASURE•Formula

where arp = the average portfolio return

arf = the average risk free rate

p= the slope of the characteristic

line during the time period

p

fpp

ararRVOL

TREYNOR MEASURE

THE CHARACTERISTIC LINEarp

p

SML

TREYNOR MEASURE

CHARACTERISTIC LINE•slope of CL

measures the relative volatility of portfolio returns in relation to returns for the aggregate market, i.e. the portfolio’s beta

the higher the slope, the more sensitive is the portfolio to the market

TREYNOR MEASURE

THE CHARACTERISTIC LINEarp

p

SML

THE SHARPE RATIO

THE REWARD TO VARIABILITY (SHARPE RATIO)•measure of risk-adjusted performance

that uses a benchmark based on the ex-post security market line

•total risk is measured by p

THE SHARPE RATIO

SHARPE RATIO•formula:

where SR = the Sharpe ratio

p = the total risk

p

fpp

ararSR

THE SHARPE RATIO

SHARPE RATIO•indicates the risk premium per unit of

total risk

•uses the Capital Market Line in its analysis

THE SHARPE RATIO

arp

p

CML

THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE BASED ON THE CAPM EQUATION

•measures the average return on the portfolio over and above that predicted by the CAPM

•given the portfolio’s beta and the average market return

])([)( RFRrERFRrE mi

THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE THE JENSEN MEASURE

•known as the portfolio’s alpha valuerecall the linear regression equation

y = + x + ealpha is the intercept

THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE DERIVATION OF ALPHA

•Let the expectations formula in terms of realized rates of return be written

•subtracting RFR from both sides

jttmtjtjt uRFRRRFRR

jttmtjtjt uRFRRRFRR

THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE DERIVATION OF ALPHA

•in this form an intercept value for the regression is not expected if all assets are in equilibrium

•in words, the risk premium earned on the jth portfolio is equal to j times a market risk premium plus a random error term

THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE DERIVATION OF ALPHA

•to measure superior portfolio performance, you must allow for an intercept

•a superior manager has a significant and positive alpha because of constant positive random errors

COMPARING MEASURES OF PERFORMANCE TREYNOR V. SHARPE

•SR measures uses as a measure of risk while Treynor uses

•SR evaluates the manager on the basis of both rate of return performance as well as diversification

COMPARING MEASURES OF PERFORMANCE

•for a completely diversified portfolioSR and Treynor give identical rankings

because total risk is really systematic variance

any difference in ranking comes directly from a difference in diversification

CRITICISM OF RISK-ADJUSTED PERFORMANCE MEASURES

Use of a market surrogateRoll: criticized any measure that

attempted to model the market portfolio with a surrogate such as the S&P500

– it is almost impossible to form a portfolio whose returns replicate those over time

– making slight changes in the surrogate may completely change performance rankings

CRITICISM OF RISK-ADJUSTED PERFORMANCE MEASURES measuring the risk free rate

using T-bills gives too low of a return making it easier for a portfolio to show superior performance

borrowing a T-bill rate is unrealistically low and produces too high a rate of return making it more difficult to show superior performance

END OF CHAPTER 24